DSP is only used in the demodulation and audio stages. If it were in the frequency synthesis stages it would be an SDR.
DSP is usually just carried out in the audio stages and can do bandwidth limits, peak and notch filtering but I haven't seen pulse noise reduction which is usually carried out in the late IF stages. There are one or two software programmes where you feed baseband audio into your computer and see the shaping that the DSP carries out - then hear the final product.